6 research outputs found
Stabilization in relation to wavenumber in HDG methods
Simulation of wave propagation through complex media relies on proper
understanding of the properties of numerical methods when the wavenumber is
real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG)
type are considered for simulating waves that satisfy the Helmholtz and Maxwell
equations. It is shown that these methods, when wrongly used, give rise to
singular systems for complex wavenumbers. A sufficient condition on the HDG
stabilization parameter for guaranteeing unique solvability of the numerical
HDG system, both for Helmholtz and Maxwell systems, is obtained for complex
wavenumbers. For real wavenumbers, results from a dispersion analysis are
presented. An asymptotic expansion of the dispersion relation, as the number of
mesh elements per wave increase, reveal that some choices of the stabilization
parameter are better than others. To summarize the findings, there are values
of the HDG stabilization parameter that will cause the HDG method to fail for
complex wavenumbers. However, this failure is remedied if the real part of the
stabilization parameter has the opposite sign of the imaginary part of the
wavenumber. When the wavenumber is real, values of the stabilization parameter
that asymptotically minimize the HDG wavenumber errors are found on the
imaginary axis. Finally, a dispersion analysis of the mixed hybrid
Raviart-Thomas method showed that its wavenumber errors are an order smaller
than those of the HDG method
A hybridizable discontinuous Galerkin method for solving nonlocal optical response models
We propose Hybridizable Discontinuous Galerkin (HDG) methods for solving the
frequency-domain Maxwell's equations coupled to the Nonlocal Hydrodynamic Drude
(NHD) and Generalized Nonlocal Optical Response (GNOR) models, which are
employed to describe the optical properties of nano-plasmonic scatterers and
waveguides. Brief derivations for both the NHD model and the GNOR model are
presented. The formulations of the HDG method are given, in which we introduce
two hybrid variables living only on the skeleton of the mesh. The local field
solutions are expressed in terms of the hybrid variables in each element. Two
conservativity conditions are globally enforced to make the problem solvable
and to guarantee the continuity of the tangential component of the electric
field and the normal component of the current density. Numerical results show
that the proposed HDG methods converge at optimal rate. We benchmark our
implementation and demonstrate that the HDG method has the potential to solve
complex nanophotonic problems.Comment: 21 pages, 8figure
Numerical investigation of a 3D hybrid high-order method for the indefinite time-harmonic Maxwell problem
Hybrid High-Order (HHO) methods are a recently developed class of methods belonging to
the broader family of Discontinuous Sketetal methods. Other well known members of the
same family are the well-established Hybridizable Discontinuous Galerkin (HDG) method,
the nonconforming Virtual Element Method (ncVEM) and the Weak Galerkin (WG) method.
HHO provides various valuable assets such as simple construction, support for fully-polyhedral
meshes and arbitrary polynomial order, great computational efficiency, physical accuracy and
straightforward support for hp-refinement. In this work we propose an HHO method for the
indefinite time-harmonic Maxwell problem and we evaluate its numerical performance. In
addition, we present the validation of the method in two different settings: a resonant cavity
with Dirichlet conditions and a parallel plate waveguide problem with a total/scattered field
decomposition and a plane-wave boundary condition. Finally, as a realistic application, we
demonstrate HHO used on the study of the return loss in a waveguide mode converter
Accuracy of Wave Speeds Computed from the DPG and HDG Methods for Electromagnetic and Acoustic Waves
We study two finite element methods for solving time-harmonic electromagnetic and acoustic problems: the discontinuous Petrov-Galerkin (DPG) method and the hybrid discontinuous Galerkin (HDG) method.
The DPG method for the Helmholtz equation is studied using a test space normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. We find that, as the parameter approaches zero, better results are obtained, under some circumstances. A dispersion analysis on the multiple interacting stencils that form the DPG method shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, its performance is compared with a standard least-squares method having a similar stencil.
We study the HDG method for complex wavenumber cases and show how the HDG stabilization parameter must be chosen in relation to the wavenumber. We show that the commonly chosen HDG stabilization parameter values can give rise to singular systems for some complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. For real wavenumbers, results from a dispersion analysis for the Helmholtz case are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method shows that its wavenumber errors are an order smaller than those of the HDG method.
We conclude by presenting some contributions to the development of software tools for using the DPG method and their application to a terahertz photonic structure. We attempt to simulate field enhancements recently observed in a novel arrangement of annular nanogaps